I want to solve below equation by relation $F(x) =\int{f(x)}$, where $F^{-1}(x)$ is $F(x)$'s inverse function.
$$f(F^{-1}(x)) = 1 - \frac{1}{f(x)}$$
$f(x)$ is probably not an elementary function. Is there a general approach to determining the features of these function? Maybe it is a fairly naive relation to get the function itself, so it's okay to get only the constraints on the function.
- Input of antiderivative is also same as input of original function, because of it is $\mathbf{R}$ to $\mathbf{R}$ function.
- Also this equation can be reduced into ordinary differential equation form,
$$f'(x) = f(x)(1-f(x))^2 f'(f^{-1}(\frac{1}{1-f(x)}))$$